A simple smoothing spline,III

Summary  Computational methods for spline smoothing are studied in the context of the linear smoothing spline. Comparisons are made between two efficient methods for computing the estimator using band-limited basis functions and the Kalman filter. In particular, the Kalman filter approach is shown to be an efficient method for computing under the Kimeldorf-Wahba representation for the estimator. Run time comparisons are made between band-limited B-spline and Kalman filter based algorithms.     

A smoothing stochastic algorithm for quantile estimation

In this paper, we provide the almost-sure convergence and the asymptotic normality of a smooth version of the Robbins–Monro algorithm for the quantile estimation. A Monte Carlo simulation study shows that our proposed method works well within the framework of a data stream.     

LOWLAD: a locally weightedL 1 smoothing spline algorithm with cross validated choice of smoothing parameters

The computation ofL ₁ smoothing splines on large data sets is often desirable, but computationally infeasible. A locally weighted, LAD smoothing spline based smoother is suggested, and preliminary results will be discussed. Specifically, one can seek smoothing splines in the spacesW _m (D), with [0, 1] ⁿ D. We assume data of the formy _i =f(t _i )+ _i ,i=1,..., N with {t _i } _i=1 ^N D, the _i are errors withE( _i )=0, andf is assumed to be inW _m . An LAD smoothing spline is the solution,s , of the following optimization problem

A class of smoothing functions for nonlinear and mixed complementarity problems

We propose a class of parametric smooth functions that approximate the fundamental plus function, (x)₊=max{0, x}, by twice integrating a probability density function. This leads to classes of smooth parametric nonlinear equation approximations of nonlinear and mixed complementarity problems (NCPs and MCPs). For any solvable NCP or MCP, existence of an arbitrarily accurate solution to the smooth nonlinear equations as well as the NCP or MCP, is established for sufficiently large value of a smoothing parameter . Newton-based algorithms are proposed for the smooth problem. For strongly monotone NCPs, global convergence and local quadratic convergence are established. For solvable monotone NCPs, each accumulation point of the proposed algorithms solves the smooth problem. Exact solutions of our smooth nonlinear equation for various values of the parameter , generate an interior path, which is different from the central path for interior point method. Computational results for 52 test problems compare favorably with these for another Newton-based method. The smooth technique is capable of solving efficiently the test problems solved by Dirkse and Ferris [6], Harker and Xiao [11] and Pang & Gabriel [28].This material is based on research supported by Air Force Office of Scientific Research Grant F49620-94-1-0036 and National Science Foundation Grant CCR-9322479.     

Possibilities of solution in stochastic decision models with recursive reward functions

A stochastic decision model with general, non-necessarily additive reward function is considered and essential properties of such reward functions are formulated which only allow a successive proceeding in the sense of dynamic programming. Conditions for recursive—additive reward functions are given which ensure the existence of optimal strategies and the usability of value iteration to find an optimal policy and the optimal total reward.     

A simple method for smoothing functions and compressing Hermite data

Let =(a=x₀<x₁<<x_n=b) be a partition of an interval [a,b] of R, and let f be a piecewise function of class C^k on [a,b] except at knots x_i where it is only of class , k_ik. We study in this paper a novel method which smooth the function f at x_i, 0in. We first define a new basis of the space of polynomials of degree 2k+1, and we describe algorithms for smoothing the function f. Then, as an application, we give a recursive computation of classical Hermite spline interpolants, and we present a method which allows us to compress Hermite data. The most part of these results are illustrated by some numerical examples. AMS subject classification 41A05, 41A15, 65D05, 65D07, 65D10     

A recursive scheme for frequency estimation using the modulating functions method

In this paper, an application of modulating functions method for estimation of the frequency of noisy sinusoids, is proposed. The unknown frequency is updated by introducing a recursive algorithm which is independent by the choice of the modulating functions type. The proposed recursive estimation formula is able to take into account possible abrupt changes or sweep in the frequency of the sinusoidal signal. The goodness of the proposed method is verified through numerical simulations.     

An algorithm for smoothing,differentiation and integration of experimental data using spline functions

This paper presents an algorithm for fitting a smoothing spline function to a set of experimental or tabulated data. The obtained spline approximation can be used for differentiation and integration of the given discrete function. Because of the ease of computation and the good conditioning properties we use normalised B-splines to represent the smoothing spline. A Fortran implementation of the algorithm is given.     

A one-parametric class of smoothing functions and an improved regularization Newton method for the NCP

In this paper, we introduce a one-parametric class of smoothing functions, which enjoys some favourable properties and includes two famous smoothing functions as special cases. Based on this class of smoothing functions, we propose a regularization Newton method for solving the non-linear complementarity problem. The main feature of the proposed method is that it uses a perturbed Newton equation to obtain the direction. This not only allows our method to have global and local quadratic convergences without strict complementarity conditions, but also makes the regularization parameter converge to zero globally Q-linearly. In addition, we use a new non-monotone line search scheme to obtain the step size. Some numerical results are reported which confirm the good theoretical properties of the proposed method.     

On Some Techniques for Streaming Data: A Case Study of Internet Packet Headers

We consider the implications of streaming data for data analysis and data mining. Streaming data are becoming widely available from a variety of sources. In our case we consider the implications arising from Internet traffic data. By implication, streaming data are unlikely to be time homogeneous so that standard statistical and data mining procedures do not necessarily apply. Because it is essentially impossible to store streaming data, we consider recursive algorithms, algorithms which are adaptive and discount the past and also algorithms that create finite pseudo-samples. We also suggest some evolutionary graphics procedures that are suitable for streaming data. We begin our discussion with a discussion of Internet traffic in order to give the reader some sense of the time and data scale and visual resolution needed for such problems.     

带Poisson 跳跃的正倒向随机延迟系统递归最优控制问题的最大值原理

本文研究了带Poisson 跳跃的正倒向随机延迟系统的递归最优控制问题. 利用经典的针状变分方法、对偶技术和带Poisson 跳跃的超前倒向随机微分方程的相关结果, 证明了最优控制的最大值原理, 包括了最优控制满足的必要条件和充分条件.     

On a class of extremal solutions of a moment problem for rational matrix‐valued functions in the nondegenerate case I

The main theme of this paper is the discussion of a parametrized family of solutions of a finite moment problem for rational matrix‐valued functions in the nondegenerate case. We will show that each member of this family is extremal in several directions concerning some point of the open unit disk. These investigations are inspired by the authors' paper [23], where a similar topic is studied in the context of the matricial Carathéodory problem. We will see that larger parts of the results presented there can be extended to the rational case studied here.The main theme of this paper is the discussion of a parametrized family of solutions of a finite moment problem for rational matrix‐valued functions in the nondegenerate case. We will show that each member of this family is extremal in several directions concerning some point of the open unit disk. These investigations are inspired by the authors' paper [23], where a similar topic is studied in the context of the matricial Carathéodory problem. We will see that larger parts of the results presented there can be extended to the rational case studied here (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A unifying theory of exactness of linear penalty functions II: parametric penalty functions

In this article, we develop a general theory of exact parametric penalty functions for constrained optimization problems. The main advantage of the method of parametric penalty functions is the fact that a parametric penalty function can be both smooth and exact unlike the standard (i.e. non-parametric) exact penalty functions that are always nonsmooth. We obtain several necessary and/or sufficient conditions for the exactness of parametric penalty functions, and for the zero duality gap property to hold true for these functions. We also prove some convergence results for the method of parametric penalty functions, and derive necessary and sufficient conditions for a parametric penalty function to not have any stationary points outside the set of feasible points of the constrained optimization problem under consideration. In the second part of the paper, we apply the general theory of exact parametric penalty functions to a class of parametric penalty functions introduced by Huyer and Neumaier, and to smoothing approximations of nonsmooth exact penalty functions. The general approach adopted in this article allowed us to unify and significantly sharpen many existing results on parametric penalty functions.     

State-space stochastic volatility models: A review of estimation algorithms

Stochastic volatility models (SVMs) represent an important framework for the analysis of financial time series data, together with ARCH-type models; but unlike the latter, the former, at least from the statistical point of view, cannot rely on the possibility of obtaining exact inference, in particular with regard to maximum likelihood estimates for the parameters of interest. For SVMs, usually only approximate results can be obtained, unless particularly sophisticated estimation strategies like exact non-gaussian filtering methods or simulation techniques are employed. In this paper we review SVM and present a new characterization for them, called ‘generalized bilinear stochastic volatility’. © 1996 John Wiley & Sons, Ltd.     

A stochastic planning framework for the discovery of complementary,agricultural systems

One of the greatest 21st century challenges is meeting the need to feed a growing world population which is expected to increase by about 35% by 2050. To meet this challenge, it is necessary to make major improvements on current food production and distribution systems capabilities, as well as to adapt these systems to expected trends such as climate change. Changing climate patterns may present opportunities for unidentified, geographical regions with adequate climate patterns to produce high-value agricultural products in a profitable and sustainable manner.This paper focuses on the design and planning aspects of a discovery process to unearth agri-food supply chains capable of generating attractive return on investments. A stochastic optimization framework is used to develop planting and harvesting schedules for a set of identified regions with complementary weather characteristics. To address the high-level of variability in the problem context, a two-stage stochastic decomposition method is used to consider a larger number of scenarios. As part of the solution process, a modeling scheme is developed that learns past interactions between entering discretized, weather scenarios and optimal first-stage solutions. In this context, machine learning and dimensionality reduction techniques are used to iteratively estimate each region's probability of belonging to first-stage solutions based on previous solution-scenario results. The implementation of the stochastic framework is shown through a case study applied to multiple locations within the US southwest states of Arizona and New Mexico.     

A limiting distribution for the number of cuts needed to isolate the root of a random recursive tree

Let X_n be the number of cuts needed to isolate the root in a random recursive tree with n vertices. We provide a weak convergence result for X_n. The basic observation for its proof is that the probability distributions of are recursively defined by , where D_n is a discrete random variable with ? , which is independent of . This distributional recursion was not studied previously in the sense of weak convergence. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

The directed and Rubinov subdifferentials of quasidifferentiable functions, Part II: Calculus

We continue the study of the directed subdifferential for quasidifferentiable functions started in [R. Baier, E. Farkhi, V. Roshchina, The directed and Rubinov subdifferentials of quasidifferentiable functions, Part I: Definition and examples (this journal)]. Calculus rules for the directed subdifferentials of sum, product, quotient, maximum and minimum of quasidifferentiable functions are derived. The relation between the Rubinov subdifferential and the subdifferentials of Clarke, Dini, Michel-Penot, and Mordukhovich is discussed. Important properties implying the claims of Ioffe’s axioms as well as necessary and sufficient optimality conditions for the directed subdifferential are obtained.     

A class of gap functions for variational inequalities

Recently Auchmuty (1989) has introduced a new class of merit functions, or optimization formulations, for variational inequalities in finite-dimensional space. We develop and generalize Auchmuty's results, and relate his class of merit functions to other works done in this field. Especially, we investigate differentiability and convexity properties, and present characterizations of the set of solutions to variational inequalities. We then present new descent algorithms for variational inequalities within this framework, including approximate solutions of the direction finding and line search problems. The new class of merit functions include the primal and dual gap functions, introduced by Zuhovickii et al. (1969a, 1969b), and the differentiable merit function recently presented by Fukushima (1992); also, the descent algorithm proposed by Fukushima is a special case from the class of descent methods developed in this paper. Through a generalization of Auchmuty's class of merit functions we extend those inherent in the works of Dafermos (1983), Cohen (1988) and Wu et al. (1991); new algorithmic equivalence results, relating these algorithm classes to each other and to Auchmuty's framework, are also given.Corresponding author.